Centrifugal Field Divergence for a Rotating Planet

Centrifugal Potential is a false potential within the context of a rotating reference frame, which can be thought of as coming from the false Centrifugal Acceleration.  I was reworking through some of the details of this and was going to post a question to Physics Stack Exchange, but it turned out that my knowledge was complete enough to not really merit a question, because I had nothing to ask, so I'll post the work here.

This has application to a rotating planet.  The mathematics for Centrifugal Potential is general to any rotating field.  I'll call it u_r.  Call the angular speed omega, and distance from the axis of rotation rho. The potential can be arrived at several ways, and they all give the following.

I would like to find the Laplacian of this.  This would be the same thing as the divergence of the Centrifugal Acceleration.  With the potential in the above form, if I understand correctly, we have to use the cylindrical version of the Laplacian.  The cords (rho,phi,z) are (distance from axis, azimuth angle, and vertical position).

The last 2 derivatives are zero, and we can expand the first.

I questioned the factor of two, but now I think that's probably correct.  This would be useful to compare this to the divergence of Newtonian gravity which can be found from Gauss' law, which I'll call u_g.

If we use Gauss' law again, then would we be able to make a 100% true statement about the integral of gravity over the surface of a rotating planet?  We would need to take dot product of gravity with the surface, but if it fits the hydrostatic condition then all gravity is normal anyway.

The two volumetric terms only depend on density.

Is it then permissible to write the following?

With this, we have two extremely interesting terms.  I think the most interesting thing is the ratio of them.  For instance, with Saturn, I calculate the rotational term is about 10% that of the gravitational term.  This means that the rotation subtracts 10% of the surface gravitational flux that its gravity creates.

It's hard to extend that statement very much further.  We would be tempted to say that someone standing on the surface of Saturn (ignoring the actual complications in doing this) would weigh 10% less due to the planet's rotation, but that's not completely correct because gravity isn't the same everywhere to begin with.

This does have some application to gravity balloons.  We can apply the concept of Gauss' Law balance to different volumes.  For instance, if we assume that the inner pressurized volume has effectively zero mass, then that implies that a rotating gravitational balloon would have a net outward field on the walls.  That means that over time things would avoid the center and move toward the walls.  That's an interesting effect and it would have a positive impact on stability, causing the Raleigh-Taylor effect to no longer want to mix the walls up.

If you wanted to take this to the absolute extreme case, you could imagine creating artificial gravity on the inside walls by rotating the structure fast enough, while still holding it together through self-gravitation.  In principle, this could possibly work, but to make Earth-like gravity, the mass required would be much greater than Earth itself.